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Given statements:

All rivers are mountain.

All forests are mountains.

Least possible Venn diagram:

Conclusion:

I. Some rivers are forests → Does not follow (There is no direct relation given between rivers and forest, so we can not say anything definite.)

II. No forest in a river → Does not follow (There is no direct relation given between rivers and forest, so we can not say anything definite.)

Hence, Either conclusion I or II follow.

According to the given information:

Given Word: REPRODUCE

There is only 'two' pair in the word 'REPRODUCE'.

Forward Pair: RU

Backward Pair: PR

Hence, the correct answer is "Two".

Given letters: R, A, L, E

The meaningful word which can formed from the given letters are,

Real → Existing or occurring in the physical world.

Rale → an abnormal crackling or rattling sound heard upon auscultation of the chest, caused by disease or congestion of the lungs.

Hence, there are "Two" meaningful words can be formed with the letters using each letter only once in each word.

Opposite surface of the open dice can be found by the below method:-

• Alternate surfaces are opposite to each other.
• No two opposite surface are touched by side or by corners.

So, by using the above rule we can find the opposite surface of open dice.

Below are the opposite surfaces:-

Note:- Two opposite faces are never seen together or adjacent to each other.

Option (1): 

A and C are opposite faces but here they are seen together which is wrong. Thus this cube cannot be made.

Option (2): 

1 and 3 are opposite faces but here they are seen together which is wrong. Thus this cube cannot be made.

Option (3): 

1 and 3 are opposite faces but here they are seen together which is wrong. Thus this cube cannot be made.

Option (4): 

None of the opposite faces are seen together nor they are adjacent to each other. Thus this cube can be made.

Hence, **"**Option (4)" is the correct answer.

Calculation:

 

Ratio of milk and water in solution K = 30.36 : 19.64

⇒ 759 ∶ 491

∴ Option (3) is the correct answer.

Given:

The average of four consecutive odd numbers is 42 less than the sum.

Formula used:

Average = Sum of observations/Number of observations

Calculation: 

Let's use 'n' to denote the smallest (first) odd number.

So, the next three consecutive odd numbers will be n + 2, n + 4, and n + 6.

According to the question,

Average = Sum of four consecutive odd numbers - 42

⇒ \(\dfrac{n+(n+2)+(n+4)+(n+6)}{4}\) = n + n + 2 + n + 4 + n + 6 - 42

⇒  \(\dfrac{4n + 12}{4}\)= 4n + 12 - 42

⇒ n + 3 = 4n - 30

⇒ 3n = 33

⇒ n = \(\dfrac{33}{3}\) = 11

∴ Number 11 comes first on this list.

Given:

Price received by pradip = 48%

Pradip spents 305 on gadgets and 55% on academics

Amount remains = Rs. 3600

Calculation:

Let the money Sana has was Rs.100

According to question he receives Rs. 48 from the pradip

Now,

30% of 48 = 14.4

55% of 48 = 26.4

Now the amount remaining = 14.4 + 26.4 = 40.8

⇒ 48 - 40.8 = 7.2

Now 7.2 Unit = Rs.3600

⇒ 1 unit = 500

So, amount that Sana had was 500 × 100 = Rs. 50000

∴ The correct answer is Rs. 50000.

Shortcut Trick

If A takes a extra days and B takes b extra days than A and B together, then time taken by A and B together is √(a × b).

Here, extra days for A (a) = 4 days.

Extra days for B (b) = 16 days.

Time taken together = √(4 × 16) = √64 = 8 days.

∴ The correct answer is 8 days.

Alternate Method

Given:

Extra time for A = 4 days

Extra time for B = 16 days

Formula Used:

If (A+B) together take x days, then A alone takes (x + a) days and B alone takes (x + b) days. The relationship is 1/(x + a) + 1/(x + b) = 1/x, which simplifies to x = √(a × b).

Calculations:

Let the time taken by A and B working jointly be x days.

⇒ Time taken by A alone = (x + 4) days

⇒ Time taken by B alone = (x + 16) days

⇒ 1/(x + 4) + 1/(x + 16) = 1/x

⇒ [(x + 16) + (x + 4)] / [(x + 4)(x + 16)] = 1/x

⇒ (2x + 20) / (x² + 20x + 64) = 1/x

⇒ x(2x + 20) = x² + 20x + 64

⇒ 2x² + 20x = x² + 20x + 64

⇒ x² = 64

⇒ x = 8 days

∴ The correct answer is 8 days.

Additional Information

Combined Efficiency Rule

If A can do a work in X days and B in Y days, then A+B together can do it in (X × Y) / (X + Y) days.

Geometric Mean Property

In time and work problems involving extra days relative to the joint time, the joint time is always the geometric mean of the extra days: x = √(a × b).

Work and Days Relation

Efficiency is inversely proportional to the number of days taken to complete the job. Efficiency = Total Work ÷ Time.

Given:

Number of heads = 50

Number of feet = 130

Calculation:

Let the number of hens be x and the number of cows is y. Therefore,

x + y = 50           ----(1)

According to the question,

2x + 4y = 130

⇒ x + 2y = 65    ----(2)

Subtracting (1) from (2), we get

y = 15

Put this value in equation (1)

⇒ x = 50 - 15 = 35

Hence, the number of hens is 35.

Calculation:

We know that if the cone is cut laterally from top the shape remaining will be the frustum with top and bottom surfaces to be spherical.

Key PointsA frustum is a 3-dimensional shape that is obtained when a smaller 3-dimensional shape is removed from the top of a larger 3-dimensional shape, keeping the top and bottom faces parallel. In other words, a frustum is a portion of a solid that remains after cutting off the top portion of the solid with a plane parallel to the base.

The most common example of a frustum is a frustum of a pyramid or cone. When the top portion of a pyramid or cone is cut off by a plane parallel to the base, the resulting shape is a frustum.

A frustum can also be formed from other 3-dimensional shapes like a frustum of a sphere (also known as a spherical frustum), but the frustum of a pyramid or cone is the most commonly encountered one.